Reference for (co)limit-preserving functor $X\mapsto R^X$I wonder what can be said in general about (say, presentable) categories $\mathcal{C}$ such that the functor $\text{set}^{op}\to\mathcal{C}$ sending $X$ to the power $I^X$ preserves finite colimits, where $I$ is the initial object. Everything I said here applies to any such category; in particular, the functor $\mathcal{C}\to\text{Set}$ corepresented by the object $I^2$ can be lifted to a right adjoint $G:\mathcal{C}\to\text{Bool}$ that in some sense provides $\mathcal{C}$ with robust notion of "idempotents".

Is there a generalization of homotopy groups to fractional dimensionsIf there were a "$1/2$-sphere" $S^{1/2}$, you would probably expect it to be a pointed space for which $S^{1/2}\wedge S^{1/2}$ is weak equivalent to $S^1$. It is easy to show (using homology, for instance) that no such space exists. This doesn't prove there is no sensible notion of $\pi_{1/2}$, but it is some evidence against it.

(A kind of) Irreducibiliy of regular open convex sets in the Cartesian spaceGiven any point $y$ in $V$, if you draw a line segment from $y$ to $x$, and then extend the line segment a little bit past $x$, you will enter the set $W\cdot U$. Thus $x$ lies on the line segment joining $y$ to some point in $W\cdot U$. Since $W\cdot U\leq C$ and $C$ is convex, $y\in C$ would imply $x\in C$.